Statistical Challenges in 21st Century Cosmology
Proceedings IAU Symposium No. 306, 2014
A. F. Heavens, J. -L. Starck & A. Krone-Martins, eds.
c International Astronomical Union 2015
doi:10.1017/S1743921314011004
How well can the evolution of the scale
factor be reconstructed by the current data?
Sandro D. P. Vitenti1 and Mariana Penna-Lima2
1
GReCO, Institut d’Astrophysique de Paris,
UMR7095 CNRS, 98 bis boulevard Arago, 75014 Paris, France
email: [email protected]
2
Divisão de Astrofı́sica, Instituto Nacional de Pesquisas Espaciais,
Av. dos Astronautas 1758, 12227-010, São José dos Campos, Brazil
email: [email protected]
Abstract. Distance measurements are currently the most powerful tool to study the expansion
history of the universe without assuming its matter content nor any theory of gravitation. In
general, the reconstruction of the scale factor derivatives, such as the deceleration parameter
q(z), is computed using two different methods: fixing the functional form of q(z), which yields
potentially biased estimates, or approximating q(z) by a piecewise nth-order polynomial function, whose variance is large. In this work, we address these two methods by reconstructing
q(z) assuming only an isotropic and homogeneous universe. For this, we approximate q(z) by
a piecewise cubic spline function and, then, we add to the likelihood function a penalty factor,
with scatter given by σr e l . This factor allows us to vary continuously between the full n knots
spline, σr e l → ∞, and a single linear function, σr e l → 0. We estimate the coefficients of q(z)
using the Monte Carlo approach, where the realizations are generated considering ΛCDM as a
fiducial model. We apply this procedure in two different cases and assuming four values of σr e l
to find the best balance between variance and bias. First, we use only the Supernova Legacy
Survey 3-year (SNLS3) sample and, in the second analysis, we combine the type Ia supernova
(SNeIa) likelihood with those of baryonic acoustic oscillations (BAO) and Hubble function measurements. In both cases we fit simultaneously q(z) and 4 nuisance parameters of the supernovae,
namely, the magnitudes M1 and M2 and the light curve parameters α and β.
Keywords. cosmology: miscellaneous, cosmology: observations, cosmology: theory
1. Introduction
The current observational data are still not able to decide between many different
models proposed to explain the recent accelerated expansion of the universe. These models include, among others, modified gravitational theories and the addition of an exotic
cosmological fluid, in the context of general relativity. As an alternative study, we can
make a pure kinematical description of this recent phase of the universe, avoiding the
choice of a gravitational theory and matter content. In this description one still models the universe as a metric manifold and uses the cosmological principle restricting the
metric to the Friedmann-Lemaı̂tre-Robertson-Walker (FLRW) metric. In this case, the
deceleration function is given by
q(z) =
(1 + z) dH(z)
− 1,
H(z) dz
(1.1)
where z is the redshift and H(z) is the Hubble function.
Different approaches have been used in the literature in order to reconstruct some
kinematic quantities such as the luminosity distance DL (z), H(z), q(z) and the jerk
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S. Vitenti & M. Penna-Lima
Figure 1. The top part of each panel shows the reconstructed curve of q(z) (spline), computed
using the mean values of the estimators, and its 1σ error bar. Each panel displays the result
f id
obtained for a given σr e l . The
√ bottom part shows the bias, q(z) − q (z), and the 1σ error bar
of the mean curve, i.e., σ/ N .
function. One of these methods assumes a priori a functional form of the scale factor
a(t) (1 + z = a0 /a), or equivalently DL (z), H(z) and so on (Riess et al. 2004, Visser
2004, Rapetti et al. 2007, Lu et al. 2011, Shafieloo 2012). This parametric reconstruction
is strongly model dependent, so, besides having small variances the results present large
biases.
In order to minimize the assumptions on the fitted curve, a second approach is to
approximate the kinematic quantity by a piecewise nth-order polynomial function. This
“non-parametric” reconstruction is dominated by the over-fitting feature, but it provides
small biased estimators (see Daly et al. 2008 and Lazkoz et al. 2012).
2. Methodology
In this work, we describe q(z) by a piecewise cubic polynomial function, also known as
cubic spline. Imposing continuous second derivatives, the only free parameters of q(z) are
its values at the knots, i.e., q(zi ) = qi , where i runs from 0 to n − 1 (being n the number
of knots). We then address both parametric and non-parametric methods including a
penalty factor Pi (σ) in the likelihood L, namely,
F = −2 ln(L) +
n
−1
Pi (σ),
(2.1)
i=2
where the penalty factor is given by
2
Pi (σ) = ((q̄i − qi )/(σabs + q̄i σr el )) ,
(2.2)
q̄i = (qi−1 + qi+1 )/2 and σabs = 10−5 . Varying the value of σr el , see Figs. 1 and 2, we
are able to recover a full n knots spline (over-fitting dominated), for large σr el , and a
single linear function in the entire redshift interval, for small σr el , where qi ’s are better
constrained but they can be biased (if the assumed functional form significantly differs
from the true one).
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Reconstruction of q(z)
221
Figure 2. Equivalent to the caption of Fig. 1. These results were obtained considering Eq. 3.1
and 12 knots.
We use the SNeIa sample from SNLS3 (Conley et al. 2011 and Sullivan et al. 2011)
and, thus, the likelihood is build as
−2 ln(LS N I a ) = Δm
T C−1
+ ln(det(CS N I a )),
S N I a Δm
(2.3)
where CS N I a = Cstat (α, β) + Csy s is the data covariance and
Δmi = 5 log10 (DL (zicm b , zihel )) − α(si − 1) + βCi + Mh i + 5 log10 (c/H0 ) + 25 − mB i . (2.4)
The SNIa phenomenological model contains four parameters α, β, M1 and M2 , where
the first two are related to the stretch-luminosity and colour-luminosity, respectively, and
M1 and M2 are absolute magnitudes.
We use data from Baryon Acoustic Oscillation (BAO) surveys such as WiggleZ, SDSS
and 6dFGRS as described in Hinshaw et al. (2013). The BAO likelihood is given by
−2 ln(LB A O ) = ΔdTv C−1
B A O Δdv ,
where CB A O is the BAO constant covariance matrix and
rr ec
.
Δdv i = dv i −
Dv (zi )
(2.5)
(2.6)
The BAO depends on the comoving sound horizon at recombination rr ec and the geometric estimate of the effective distance Dv (zi ). The last is calculated directly rewriting
the distance as an integral of q(z), however, rr ec have to be treated as an additional
parameter since our model does not define rr ec .
The Hubble function likelihood is build as
T C−1
−2 ln(LH u bble ) = ΔH
H u bble ΔH,
(2.7)
where CH u bble is the diagonal constant covariance matrix and ΔHi = Hi − H(zi ). This
likelihood includes the H(z) measurements which were obtained independently of BAO.
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S. Vitenti & M. Penna-Lima
The data is summarize, for example, in Zheng et al. (2014). We also include the H0
(z = 0) measurement from Riess et al. (2011).
We apply the Monte Carlo (MC) approach to obtain the estimators of qi ’s, α, β, M1 and
M2 . For this, we first define a fiducial model from which N random realizations will be
generated. In particular, we assume a flat universe (k = 0) and a ΛCDM cosmological
model with H0 = 73 km s−1 Mpc−1 , Ωm = 0.3 and ΩΛ = 0.7. For each realization,
the best-fitting values of the parameters are obtained by minimizing Eq. (2.1). At each
step, i.e., for each resample, the arithmetic mean and the variance of the estimators
are computed. This loop ends when the required precision is achieved. In this work, the
number of realizations needed varies between 104 and 2 × 105 .
3. Results and Concluding Remarks
We first apply the methodology considering a spline of q(z) with 8 knots within the
redshift interval [0.01, 1.4]. We perform this analysis using only the SNLS3 data, i.e., its
covariance matrix to generate the realizations, and Eqs. (2.5) and (2.1) to obtain the
means and variances of the 9 qi estimators, α, β, M1 and M2 . We consider 4 different
values of σr el = 0.05, 0.1, 0.5 and 0.8. In Fig. 1 we show the reconstructed q(z) function,
computed using the mean of the estimators, and its 1σ error bar.
In this second case we consider 12 knots within the redshift interval [0, 2.3]. We perform
this analysis using the joint likelihood:
−2 ln(L) = −2 ln(LS N I a ) − 2 ln(LB A O ) − 2 ln(LH u bble ).
(3.1)
Fig. 2 displays the results for those 4 different σr el values.
It is worth emphasizing that the parameters α, β, M1 , M2 and H0 are recovered in all
cases, i.e. ML estimators for these quantities have negligible biases.
In both figures we plotted the threshold line, q(z) = 0, which indicates the deceleration/acceleration regions. In the 8 cases studied, we obtain the indication of an accelerated
expansion with more than 1σ confidence level in the interval 0 z 0.4.
In this work, we used ΛCDM as the fiducial model, in which q(z) is almost linear.
Therefore, in order to obtain a more conservative reconstruction of q(z), it is necessary to
apply this study for fiducial models with different functional forms of q(z). Once we get
suitable number of knots and σr el , which work well for all fiducial models, we can apply
the method using real data and recover a conservative estimate of the recent expansion
history of the universe (Vitenti & Penna-Lima, in prep.).
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at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921314011004